Arbelos

In geometry, an arbelos is a plane region bounded by three semicircles connected at the corners, all on the same side of a straight line (the baseline) that contains their diameters.

The earliest known reference to this figure is in the Book of Lemmas by Archimedes, where some of its mathematical properties are stated as Propositions 4 through 8.

Etymology

The name arbelos, used by Archimedes, comes from Greek ἡ ἄρβηλος he árbēlos or ἄρβυλος árbylos, meaning "shoemaker's knife", a knife used by cobblers from antiquity to the current day, whose blade is said to resemble the geometric figure.

Properties

Two of the semicircles are necessarily concave, with arbitrary diameters a and b; the third semicircle is convex, with diameter a+b.

In the following sections, the corners of the arbelos are labeled , , and , such that the diameter of the outer semicircle is , assumed to have unit length; and the diameters of the inner semicircles are and , assumed to have lengths r and 1−r, respectively. The letter denotes the point where the outer semicircle intercepts the line that is perpendicular to the diameter through the point .